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Pre-Trained Low-Rank Tensor Decomposition for Multi-Dimensional Image Recovery

Bing-Zhang Fu, Zhi-Long Han, Ting-Zhu Huang, Xi-Le Zhao, Deyu Meng

arXiv:2609.12843v1cs.CV

TL;DR

Existing tensor decomposition methods neglect common structure across images, while deep variants can require costly training and many parameters. PLTD combines a frozen large-vision-model prior with a learnable low-rank component and transform; the paper reports strong recovery across benchmarks and image domains.

  • Problem

    Existing deep tensor decomposition methods neglect shared image structure and typically require costly training and many parameters to learn each target from scratch.

    Source: §I · p. 2

  • Method

    PLTD combines a frozen DINOv3-derived common latent structure, a learnable low-rank instance-specific term, and a lightweight transform.

    Source: §I · p. 2, A. The Proposed PLTD · p. 5

  • Results

    PLTD achieves state-of-the-art recovery across multiple benchmark datasets and generalizes across challenging cross-domain and real-world images.

    Source: §I · p. 2

  • Takeaways & Limitations

    Across color, multispectral, remote-sensing, and magnetic-resonance images, PLTD is reported to deliver superior recovery with fewer learnable parameters and a smaller carbon footprint.

    Source: §VII · p. 13

Abstract

from arXiv · show

Recently, tensor decompositions are prevalent for multi-dimensional image representation, which learn the instance-specific structure of each image from scratch. However, tensor decompositions neglect the common structure across different images, leading to limited semantic modeling capability, high computational cost, and a large number of learnable parameters. To address this challenge, we suggest the first pre-trained low-rank tensor decomposition (PLTD) framework, which organically integrates the pre-trained large vision model into the classical tensor decomposition framework. Beyond the shallow and untrained deep tensor decomposition, the suggested PLTD achieves an unprecedented balance among higher recovery fidelity, fewer learnable parameters, and smaller carbon footprint. Specifically, PLTD factorizes the target tensor into a latent tensor and a learnable transform that maps the latent tensor back to the original data domain. The latent tensor consists of two indispensable and complementary terms, i.e., a fixed pre-trained latent tensor and a learnable low-rank latent tensor. The fixed pre-trained latent tensor is distilled from a pre-trained large vision model (i.e., DINOv3) to capture the common structure of the target tensor, while the learnable low-rank latent tensor characterizes the instance-specific structure of the target tensor. To examine the potential of PLTD, we develop the corresponding multi-dimensional image recovery model and theoretically justify the advantages of this framework. Additionally, we discuss the connections between PLTD and classical tensor decomposition frameworks. Extensive experiments on multi-dimensional image recovery demonstrate that PLTD consistently achieves superior performance compared with state-of-the-art methods.

I. INTRODUCTION

Multi-dimensional image recovery needs models that exploit structural redundancy, yet shallow and untrained deep tensor decompositions leave common structure across images unused. PLTD introduces pre-training into tensor decomposition, and the paper reports strong recovery performance across benchmarks and image domains.

A. Shallow Tensor Decomposition Methods

B. Untrained Deep Tensor Decomposition Methods

I. INTRODUCTION

  • PLTD combines a frozen DINOv3-derived common structure with a learnable low-rank instance-specific structure and a lightweight transform.Source: §I · p. 2
  • The paper presents PLTD as a balance of recovery fidelity, parameter count, and carbon footprint, and also develops a recovery model with theoretical justification and links to classical decompositions.Source: §I · p. 2
  • PLTD achieves state-of-the-art recovery across multiple benchmark datasets and generalizes across challenging cross-domain and real-world images.Source: §I · p. 2

III. NOTATION AND PRELIMINARIES

  • The preliminaries define tensor notation and operations alongside CP, Tucker, and block-term decomposition conventions.Source: §III · p. 3

IV. PROPOSED METHOD

The proposed framework combines a pre-trained latent tensor with a learnable low-rank tensor and transform. It motivates PLTD against untrained deep decomposition and presents a recovery model with an error bound.

IV. PROPOSED METHOD

  • The section introduces PLTD and relates it to classical shallow and untrained deep tensor decomposition paradigms.Source: §IV · p. 3
  • The proposed multi-dimensional image recovery model is accompanied by a theoretical recovery-error bound.Source: §IV · p. 3
  • PLTD combines a pre-trained latent tensor and a learnable low-rank tensor, unlike untrained deep methods that learn a heavy network from scratch.Source: Figure 2 · p. 4

A. The Proposed PLTD

PLTD represents a target tensor through a latent tensor and a learnable transform. Its latent representation combines a frozen DINOv3 prior for shared structure with a learnable low-rank term for target-specific structure.

A. The Proposed PLTD

  • PLTD factorizes the target tensor into a latent tensor and a lightweight transform, with the latent tensor combining pre-trained and learnable low-rank terms.Source: A. The Proposed PLTD · p. 4
  • The pre-trained latent tensor is extracted from a coarse interpolated observation using frozen DINOv3, supplying common structural and semantic information.Source: A. The Proposed PLTD · p. 5
  • The learnable low-rank term supplements the pre-trained tensor's dominant structure with target-specific structure to align the latent representation.Source: A. The Proposed PLTD · p. 5, Figure 3 · p. 5
  • A lightweight CNN transform models dependencies among frontal slices and maps the latent tensor to the recovery tensor with few learnable parameters.Source: A. The Proposed PLTD · p. 5
  • PLTD is presented as balancing fidelity, parameter count, and carbon footprint; its low-rank formulation includes slice-wise matrix, CP, Tucker, and block-term decompositions as special cases.Source: A. The Proposed PLTD · pp. 4–5

B. PLTD-based Multi-Dimensional Image Recovery Model

The recovery model combines a fixed pre-trained latent tensor with learnable low-rank factors and a lightweight transform. Its theoretical analysis relates recovery error to model complexity and finds a tighter bound than untrained deep tensor decomposition under general conditions.

B. PLTD-based Multi-Dimensional Image Recovery Model

  • PLTD combines a fixed pre-trained latent tensor with learnable low-rank factors, then uses a lightweight transform to map the latent tensor to the original domain.The observation model recovers X from incomplete observations O while keeping the pre-trained tensor S frozen.Source: B. PLTD-based Multi-Dimensional Image… · p. 5
  • Adam optimizes the transform parameters and low-rank factors, and the recovered tensor is obtained from the optimized model.The algorithm uses a maximum iteration count as its stopping criterion.Source: B. PLTD-based Multi-Dimensional Image… · p. 6

C. Theoretical Error Bound

  • Under general conditions, introducing pre-trained common structure yields a tighter recovery error bound than untrained deep tensor decomposition.The theorem’s assumptions include random sampling and bounded tensor entries and model outputs.Source: C. Theoretical Error Bound · p. 6
  • The bound depends on effective model complexity C = r(n1 + n2) + ˆn3 + Hg, combining latent-tensor complexity with transform complexity.Hg denotes the number of learnable parameters in the convolutional network.Source: C. Theoretical Error Bound · p. 6

V. EXPERIMENTS

Experiments compare PLTD with shallow and untrained deep tensor decomposition methods across multiple image-recovery settings. PLTD achieves strong quantitative and visual recovery results on color images and MSIs, including under structural missing.

V. EXPERIMENTS

  • Experiments compare PLTD with shallow and untrained deep methods across color, MSI, remote-sensing, and MRI recovery.Remote-sensing metrics appear in Table III, and MRI inpainting results are shown at 15% SR.Source: §V · p. 6, Table III · p. 9, Figure 8 · p. 10

B. Multispectral Image Recovery

A. Color Image Recovery

  • At 15% SR, Butterfly PSNR rises from 20.591 dB to 25.256 dB, and PLTD leads on most color-image sampling rates.The comparison reports clear advantages over MTTD and DELTA across the tested color images.Source: A. Color Image Recovery · pp. 7–8, Table I · p. 7

B. Multispectral Image Recovery

C. Remote Sensing Image Recovery

Across remote-sensing and MRI recovery, PLTD generalizes beyond color imagery, while the discussion attributes its modeling and efficiency benefits to combining shared pre-trained structure with learnable low-rank structure.

C. Remote Sensing Image Recovery

  • On four-band, 0.6 m NAIP CNIR images, PLTD achieves higher PSNR and SSIM across sampling rates and recovers clearer textures and field boundaries than traditional tensor methods.The authors describe this as evidence of generalization to real-world remote-sensing images.Source: C. Remote Sensing Image Recovery · p. 10

D. Magnetic Resonance Imaging Recovery

  • At 10% sampling, PLTD reaches 20.608 dB PSNR and 0.809 SSIM, exceeding second-best results by 1.097 dB and 0.202, respectively.Across random-missing MRI experiments at 10%, 15%, and 20% sampling, it also has the highest PSNR and SSIM at each rate; visual results show clearer tissue boundaries.Source: D. Magnetic Resonance Imaging Recovery · pp. 10–11

A. Benefits of Pre-Trained Common Structure

  • Compared with classical instance-specific tensor modeling, PLTD reports better recovery performance with significantly fewer learnable parameters and shorter running time.The discussion frames fixed pre-trained common structure plus lightweight low-rank instance-specific modeling as the basis for this trade-off.Source: A. Benefits of Pre-Trained Common… · p. 11
  • The discussion proposes using common structure as a foundation while retaining low-rank decomposition to model each target tensor's instance-specific structure.The authors present this shift as a direction for more expressive and efficient tensor-decomposition frameworks.Source: A. Benefits of Pre-Trained Common… · p. 11

B. Synergy between Pre-Trained Latent Tensor and Learnable Low-Rank Latent Tensor

  • Using only the low-rank latent tensor gives inferior recovery, while using only the pre-trained tensor gives limited performance; integrating both achieves the best results.The pre-trained tensor supplies common structure, and the learnable low-rank tensor complements it with instance-specific detail.Source: B. Synergy between Pre-Trained Latent… · p. 11, Figure 9 · p. 11

C. Other Low-Rank Structures in PLTD

PLTD combines pre-trained semantic structure with learnable low-rank modeling, and the paper reports recovery, downstream detection, and early-optimization benefits alongside a favorable efficiency–fidelity balance.

C. Other Low-Rank Structures in PLTD

  • CP, Tucker, and slice-wise matrix factorization perform comparably in PLTD, while the adopted low-rank structure has the highest PSNR and SSIM with fewer parameters and shorter running time.These comparisons support flexibility across low-rank representations and the selected structure's recovery–efficiency trade-off.Source: C. Other Low-Rank Structures in PLTD · pp. 11–12, Table VII · p. 12

D. High-Level Vision Task

  • With pre-trained semantic knowledge, YOLOv11 produces more reliable detections with higher confidence; without it, examples include a missed elephant and an elephant misclassified as a cow.The authors say semantic knowledge helps preserve compatibility with high-level vision tasks while retaining pixel-level fidelity.Source: D. High-Level Vision Task · p. 12, Figure 10 · p. 12

E. Convergence Behavior Analysis

VII. CONCLUSION

  • Across color, multispectral, remote-sensing, and MRI images, PLTD consistently achieves superior recovery with significantly fewer learnable parameters and a smaller carbon footprint.The conclusion presents transferable common structure from pre-trained vision models, combined with compact low-rank modeling, as a promising direction for tensor decomposition.Source: §VII · p. 13

APPENDIX A PROOF OF THEOREM

The appendix establishes a recovery-error bound for the PLTD model class using covering numbers and a random-observation sampling argument, under stated parameter, architecture, and activation assumptions.

APPENDIX A PROOF OF THEOREM

  • The proof derives a recovery-error bound by covering the model class and applying a random-observation sampling lemma under the theorem's stated assumptions.The theorem concerns recovery from randomly sampled entries and gives a probability bound of at least 1 − 2N^-1.Source: Appendix A · pp. 15, 17
  • The covering argument assumes bounded factor and network parameters, bounded intermediate tensors, and an η-Lipschitz activation, enabling layerwise control of approximation error.The appendix propagates errors through the network using these assumptions.Source: Appendix A · pp. 15–16
  • The analyzed model class combines fixed pre-trained tensor S with the low-rank tensor (AB) ◦c and maps the result through learnable transform gθ.The transform is specified as an L-layer convolutional neural network.Source: Appendix A · pp. 15, 17
  • Because S is fixed, it adds no parameter dimension to the covering-number exponent; the low-rank component uses r(n1 + n2) + n̂3 scalar parameters.The discussion contrasts this with untrained deep tensor methods, whose latent-tensor modeling can require many more learnable parameters.Source: Appendix A · p. 17
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